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Necessary and Sufficient Quantum Information Characterization Of Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering Marco Piani1, 2 and John Watrous3, 4 1Institute for Quantum Computing & Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada 2Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, UK 3Institute for Quantum Computing & School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada 4Canadian Institute for Advanced Research, 180 Dundas Street West, Suite 1400, Toronto, Ontario M5G 1Z8, Canada Steering is the entanglement-based quantum effect that embodies the \spooky action at a dis- tance" disliked by Einstein and scrutinized by Einstein, Podolsky, and Rosen. Here we provide a necessary and sufficient characterization of steering, based on a quantum information processing task: the discrimination of branches in a quantum evolution, which we dub subchannel discrimina- tion. We prove that, for any bipartite steerable state, there are instances of the quantum subchannel discrimination problem for which this state allows a correct discrimination with strictly higher probability than in the absence of entanglement, even when measurements are restricted to local measurements aided by one-way communication. On the other hand, unsteerable states are useless in such conditions, even when entangled. We also prove that the above steering advantage can be exactly quantified in terms of the steering robustness, which is a natural measure of the steerability exhibited by the state. PACS numbers: 03.67.Mn, 03.67.Bg, 03.65.Ud Entanglement is a property of distributed quantum for?" can arguably be considered limited [9, 12]. Further- systems that does not have a classical counterpart and more, the quantification of steering has just started to be challenges our everyday-life intuition about the physi- addressed [24]. cal world [1]. It is also the key element in many quan- In this Letter we fully characterize and quantify steer- tum information processing tasks [2]. The strongest fea- ing in an operational way that mirrors the asymmetric ture exhibited by entangled systems is non-locality [3]. features of steering, and that breaks new ground in the A weaker feature related to entanglement is steering: investigation of the usefulness of steering. We prove that roughly speaking, in quantum steering one party can in- every steerable state is a resource in a quantum infor- duce very different ensembles for the local state of the mation task that we dub subchannel discrimination, in other party, beyond what is possible based only on a con- a practically relevant scenario where measurements can ceivable classical knowledge about the other party's \hid- only be performed locally. Subchannel discrimination is den state" [4, 5]. Steering embodies the \spooky action at the identification of which branch of a quantum evolu- a distance"|in the words of Einstein [6]|identified by tion a quantum system undergoes (see Fig. 1). It is well Schr¨odinger[7], scrutinized by Einstein, Podolsky, and known that entanglement between a probe and an an- Rosen [8], and formally put on sound ground in [4, 5]. cilla can help in discriminating different channels [31{44]. Not all entangled states are steerable, and not all steer- In [45] it was proven that every entangled state is useful able states exhibit nonlocality [4, 5], but states that ex- in some instance of the subchannel discrimination prob- hibit steering allow for the verification of their entangle- lem. Ref. [46] analyzed the question of whether such an ment in a semi-device independent way: there is no need advantage is preserved when joint measurements on the to trust the devices used by the steering party [4, 5, 9]. output probe and the ancilla are not possible. Here we Besides its foundational interest, steering is interesting prove that, when only local measurements coordinated in practice in bipartite tasks, like quantum key distribu- by forward classical communication are possible, every tion (QKD) [10], where it is convenient or appropriate steerable state remains useful, while non-steerable en- to trust the devices of one of two parties, but not neces- tangled states become useless. We further prove that this sarily of the other one. For example, by exploiting steer- usefulness, optimized over all instances of the subchannel ing, key rates unachievable in a fully device-independent discrimination problem, is exactly equal to the robustness approach [11] are possible, still assuming less about the of steering, a natural way of quantifying steering using devices than in a standard QKD approach [12]. For these techniques similar to the ones used in [24], but based on reasons, steering has recently attracted significant in- the notion of robustness [47{50]. terest, both theoretically and experimentally [13{30], Preliminaries: entanglement and steering.| We will mostly directed to the verification of steering. Nonethe- denote using a ^ (hat) mathematical entities that are less, an answer to the question \What is steering useful \normalized." So, for example, a positive semidefinite op- 2 en- erator with unit trace is a (normalized) stateρ ^. An Λ1[ρ] Λ1 semble E = fρaga for a stateρ ^ is a collection of substates ρ ≤ ρ^ such that P ρ =ρ ^. Each substate ρ is pro- a a a a Λ2 Λ2[ρ] portional to a normalized stateρ ^a, ρa = paρ^a, with . ρ . pa = Tr(ρa) the probability ofρ ^a in the ensemble. An as- . Λa semblage A = fExgx = fρa xga;x is a collection of ensem- j P . Λ [ρ] bles Ex for the same stateρ ^, one for each x, i.e., a ρa x = . a j Λˆ . ρ^, for all x. For example, E = f 1 j0ih0j; 1 j1ih1jg and . ˆ 2 2 p . Λ[ρ] 1 1 E0 = f 2 j+ih+j; 2 |−ih−|}, with |±i := (j0i ± j1i)= 2, are both ensembles for the maximally mixed state 11=2 FIG. 1. A decomposition of a channel into subchannels can be of a qubit, and taken together they form an assemblage seen as a decomposition of a quantum evolution into branches ^ A = fE; E0g for 11=2. Along similar lines, a measurement of the evolution. If fΛaga is an instrument for Λ, then we can ^ assemblage MA = fMa xga;x is a collection of positive imagine that the evolution ρ 7! Λ[ρ] has branches ρ 7! Λa[ρ], j P where each branch takes place with probability Tr(Λa[ρ]). The operators Ma x ≥ 0 satisfying a Ma x = 11 for each x, j j ^ which thus represents one positive-operator-valued mea- transformation described by the total channel Λ can be seen sure (or POVM), describing a quantum measurement, for as the situation where the \which-branch" information is lost. An example of subchannel discrimination problem is that of each x. For a fixed bipartite stateρ ^AB, every measure- distinguishing between the two quantum evolutions Λ [ρ] = p i ment assemblage on Alice leads to an assemblage on Bob K ρKy, i = 0; 1, with K = j0ih0j + 1 − γj1ih1j and K = pi i 0 1 via γj0ih1j, corresponding to the so-called amplitude damping ^ B A channel Λ = Λ0 + Λ1 [2]. ρa x = TrA Ma xρ^AB : (1) j j On the other hand, every assemblage on Bob fσa xga;x sep P j one has TrA(Ma xσAB) = λ p(λ)p(ajx; λ)σB(λ), with has a quantum realization (1) for someρ ^AB satisfying j p(ajx; λ) = TrA(Ma xσA(λ)). It follows that entangle- P j ρ^B = TrA(^ρAB) = x σa x =:σ ^B and for some measure- j ment is a necessary condition for steerability, and, in ment assemblage [51]. turn, a steerable assemblage is a clear signature of en- An assemblage A = fρa xga;x is unsteerable if j tanglement. Interestingly, not all entangled states lead to steerable assemblages by the action of appropriate local US X X ρa x = p(λ)p(ajx; λ)^σ(λ) = p(ajx; λ)σ(λ); (2) measurement assemblages [4, 5]; we call steerable states j λ λ those that do, and unsteerable states those that do not. There exist entangled states that are steerable by one for all a; x, for some probability distribution p(λ), condi- party but not the other (see, e.g., [22]). In this Letter, tional probability distributions p(ajx; λ), and statesσ ^(λ). when we refer to a state being steerable or unsteerable, Here λ indicates a (hidden) classical random variable, it is always to be assumed that Alice is the steering party. and we also introduced subnormalized states σ(λ) = Channel and subchannel identification.| A subchannel p(λ)^σ(λ). We observe that every conditional probability Λ is a linear completely positive map that is trace non- distribution p(ajx; λ) can be written as a convex combi- increasing: Tr(Λ[ρ]) ≤ Tr(ρ), for all states ρ. If a sub- nation of deterministic conditional probability distribu- channel Λ is trace-preserving, Tr(Λ[ρ]) = Tr(ρ), for all tions: p(ajx; λ) = P p(νjλ)D(ajx; ν), where D(ajx; ν) = ν ρ, we use the ^notation and say that Λ^ is a channel. An δa;fν (x) is a deterministic response function labeled by ν. instrument I = fΛaga for a channel Λ^ is a collection of This means that, by a suitable relabeling, ^ P subchannels Λa such that Λ = a Λa (see Figure 1). Ev- US X ery instrument has a physical realization, where the index ρa x = D(ajx; λ)σ(λ) 8a; x; (3) j a can be considered available to some party [2, 53, 54]. λ Fix an instrument fΛaga for a channel Λ,^ and con- where the summation is over labels of deterministic re- sider a measurement fQbgb on the output space of Λ.^ The sponse functions.
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